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Journal of Lie Theory 26 (2016), No. 4, 1107--1144
Copyright Heldermann Verlag 2016

The Tame Butcher Group

Geir Bogfjellmo
NTNU Trondheim, Alfred Getz' vei 1, 7491 Trondheim, Norway

Alexander Schmeding
NTNU Trondheim, Alfred Getz' vei 1, 7491 Trondheim, Norway

The Butcher group is a powerful tool for analysing integration methods for ordinary differential equations, in particular Runge-Kutta methods. Recently, a natural Lie group structure has been constructed for this group. Unfortunately, the associated topology is too coarse for some applications in numerical analysis. In the present paper, we propose to remedy this problem by replacing the Butcher group with the subgroup of all exponentially bounded elements. This "tame Butcher group" turns out to be an infinite-dimensional Lie group with respect to a finer topology. As a first application, we show that the correspondence of elements in the tame Butcher group with their associated B-series induces certain Lie group (anti)morphisms.

Keywords: Butcher group, infinite-dimensional Lie group, Silva space, regularity of Lie groups, B-series, group of germs of diffeomorphisms.

MSC: 22E65; 65L06, 58A07

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